The finance industry uses a number of longstanding models to calculate risk, products and investment strategies. Prominent models include the modern portfolio theory (MPT) first put forward by economist Harry Markowitz, the arbitrage pricing theory (APT), the capital asset pricing model (CAPM) and the single index model (SIM).
But these prominent models are regularly called into question, especially when major stock market shakeups occur. The global financial crisis of 2008 was one such event, and it caught many traders unawares. But less significant market developments, like the Swiss franc shock on January 15, 2015, have shed light on the limits of longstanding financial theories.
One of the most prominent critics of the modern portfolio theory and its close relative the “standard finance theory”, was renowned mathematician Benoît Mandelbrot (1924-2010). The founder of fractal geometry was also interested in phenomena occurring outside of the mathematics space, and applied his new instruments to many of these. He spent a great deal of time analyzing financial markets.
The standard model called into question
Mandelbrot did not have an issue with the math behind the standard finance theory (comprised of the modern portfolio theory and its key developments), but he did see flaws in its underlying assumptions. Standard finance theory assumes the best-case scenario with regards to investor and market behavior. Mandelbrot believed that markets are turbulent, insecure and cannot be pressed into the mold of standard finance theory. Here we list his most important objections to the standard finance theory.
1. Investors often behave irrationally. Standard models assume that investors will trade rationally and (primarily) that investors always look to maximize their own gains. Mandelbrot believed that irrational behavior and moves not driven by a desire to make a profit must be accounted for because they have a direct influence on markets.
2. Changes in rates do not follow normal distribution patterns. The modern portfolio theory and other theories which build on it, such as the capital asset pricing model (CAPM), assume the normal distribution (also called Gaussian distribution) of share profits. The normal distribution pattern assumes that risk follows a Gaussian bell curve, which means that extreme fluctuations in market rates should occur very rarely.
Mandelbrot, on the other hand, believed that the “mild randomness” assumed by this theory did not accurately match the wild randomness of financial markets.
If normal distribution patterns were in fact the norm, a loss or gain of more than 7 percent on the Dow Jones stock exchange would happen once every 300,000 years. Of course this has not been the actual case, with more than 48 separate occurrences of 7 percent fluctuations within one day recorded between 1990 and 1997.
3. Rate changes affect each other. Standard finance theory assumes a perfect capital market. In a perfect world, all of the information relevant to a share would be figured into its price. Past rate changes would not have any effect on current rate changes.
According to Mandelbrot, many financial rate tables have a memory, in that current rates have a direct effect on future rates. A large number of rate drops occurring over a certain time-frame heightens the risk of further drops in the time-frame that follows. So the current direction of market rates becomes less important than market volatility based on the frequency and scale of rate fluctuations.
So what exactly is a fractal?
A fractal is a pattern or shape of which each individual part is a reflection of the whole. So, as a rule, the shape of every part of a fractal structure is identical to the shape of the structure as a whole, and vice-versa.
Fractal-like structures can also be found in nature. The structures of coastlines and mountain ranges, for example, can often be replicated using fractal geometry. Similarly, astoundingly realistic simulations of stock market rates can be achieved using fractal patterns. Fractal patterns in which the sizes of different areas change in varying ways are known as multifractal.
Mandelbrot developed computer-assisted multifractal models that could simulate and predict rate changes of various stocks. Interestingly, stock market patterns show very little variation when scaled up or down. In other words, the same pattern shows regardless of the time frame over which you observe rates.
Throughout his analyses, Mandelbrot was not looking to find the exact cause of rate fluctuations. A good analogy can be made based on a motor car: You don’t need to know everything about how an automobile works in order to drive it. Or in the case of stock trading: You don’t need to know exactly why markets behave the way they do in order to trade successfully. Understanding that stock market rates regularly behave irregularly is a good start to understanding market rates.
We’ve compiled 5 of the most important practical discoveries of Mandelbrot’s financial theories here:
1. Turbulence is the norm
Financial experts normally recommend that investors diversify their stock market investments, hold onto their stocks over a long period of time and focus on average rates rather than temporary fluctuations. But according to Mandelbrot, this strategy is not completely suited to the turbulent nature of the market, which is heavily influenced by single, major events. Successful investors often make large gains on short-term investments rather than on long-term trading strategies.
2. Rates Can Jump
People often look for continuity in processes and events, even where it doesn’t exist. Prevailing economic models too are primarily based on the assumption of continuity and “smooth” market transitions. But these theories don’t match the radical hikes and drops we see in the stock market, and therefore, according to Mandelbrot, are not correct.
Even seemingly insignificant reports or mood swings can lead to major changes in market rates. The continuous flood of information circulated via the Internet has greatly heightened these outbreaks.
3. Trading cycles are relative
Popular theories like the capital asset pricing model are built on the assumption that there is an “average” investor. They don’t account for the fact that different investors hold onto stocks for different amounts of time. Mandelbrot’s model distributes trading cycles differently to the way it is done using standard models.
4. Market analytics are often irrelevant
Mandelbrot claimed that long term dependencies connect most changes in market rates. This does create a tendency towards rate fluctuations within a specific range, but rates cannot be charted using set, non-fractal patterns at a rate level. Thus, the predictions made by “chartists” are best taken with a grain of salt. Even the high and low cycles explained by the Kondratiev wave theory may also be the product of chance.
5. Volatility is not a matter of chance
Mandelbrot believed that volatility and risk can be calculated to some extent, even though actual rates are unpredictable because upcoming rate changes cannot be based on previous stock market climbs. But there is a tendency towards major changes in stock market rates following a sudden rate hike. Volatility predictions can be compared with weather predictions in that they are based on a loose pattern.
Can fractals help you trade successfully?
Mandelbrot’s discovery of fractals has resonated throughout financial circles. A number of investment funds already use Mandelbrot’s theories as their model for stock market investment.
But fractal financial and economic analysis is still a new science. Nobody can say for sure whether or not fractal patterns will someday lead to optimal trading portfolios churning out above-average profits.
In the meantime, there’s nothing to stop you from making use of fractal-based volatility predictions. Traders who bet on market volatility through options or other investment vehicles should theoretically get more accurate predictions using fractal patterns rather than standard models.
Volatility indexes based on fractal models should be able to predict stock market crashes over the short-term. Much like a weather report, though, the further ahead of the actual event a prediction is made, the less reliable it will be.
Conclusion: Fractal models are not the magic key to beating the market. But in the best case they can help you recognize upcoming high-volatility periods and thus avoid making risky investments and incurring heavy losses.
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Capital Asset Pricing Model (CAPM)
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